Symbolic Computation Software Research Articles

Assorted Spatial Optical Dynamics of a Generalized Fractional Quadruple Nematic Liquid Crystal System in Non-Local Media

Nematicons upgrade the recognition of light localization in the reorientation of non-local media. the current research employs a powerful integral scheme using a different procedure, namely, the modified simple equation method (MSEM), to analyze nematicons in liquid crystals from the controlling model. The expanded MSEM is investigated to enlarge the applicability of the standard one. The suggested expansion depends on merging the MSEM and the ansatz method. The new generalized nonlinear n-times quadruple power law is included. With the aid of the symbolic computational package Mathematica, new explicit complex hyperbolic, periodic, and more exact spatial soliton solutions are derived. Moreover, the related existence constraints are obtained. To show the dynamical properties of some of the obtained nematicons, three-dimensional profiles with corresponding contours are depicted with the choice of appropriate values of arbitrary parameters. The fractional impacts in various applicable senses are analyzed to investigate the generality of the considered model.

Numerical Study of Time-Fractional Schrödinger Model in One-Dimensional Space Arising in Mathematical Physics

This study provides an innovative and attractive analytical strategy to examine the numerical solution for the time-fractional Schrödinger equation (SE) in the sense of Caputo fractional operator. In this research, we present the Elzaki transform residual power series method (ET-RPSM), which combines the Elzaki transform (ET) with the residual power series method (RPSM). This strategy has the advantage of requiring only the premise of limiting at zero for determining the coefficients of the series, and it uses symbolic computation software to perform the least number of calculations. The results obtained through the considered method are in the form of a series solution and converge rapidly. These outcomes closely match the precise results and are discussed through graphical structures to express the physical representation of the considered equation. The results showed that the suggested strategy is a straightforward, suitable, and practical tool for solving and comprehending a wide range of nonlinear physical models.

Numerical model of Phobos’ motion incorporating the effects of free rotation

Context. High-precision ephemerides are not only useful in supporting space missions, but also in investigating the physical nature of celestial bodies. This paper reports an update to the orbit and rotation model of the Martian moon Phobos. In contrast to earlier numerical models, this paper details a dynamical model that fully considers the rotation of Phobos. Here, Phobos’ rotation is first described by Euler’s rotational equations and integrated simultaneously with the orbital motion equations. We discuss this dynamical model, along with the differences with respect to the model now in use. Aims. This work is aimed at updating the physical model embedded in the ephemerides of Martian moons, considering improvements offered by exploiting high-precision observations expected from future missions (e.g., Japanese Martian Moons exploration, MMX), which fully supports future studies of the Martian moons. Methods. The rotational motion of Phobos can be expressed by Euler’s rotational equations and integrated in parallel with the equations of the orbital motion of Phobos around Mars. In order to investigate the differences between the two models, we first reproduced and simulated the dynamical model that is now used in the ephemerides, but based on our own parameters. We then fit the model to the newest Phobos ephemeris published by Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE). Based on our derived variational equations, the influence of the gravity field, the Love number, k2, and the rotation behavior were studied by fitting the full model to the simulated simple model. Our revised dynamic model for Phobos was constructed as a general method that can be extended with appropriate corrections (mainly rotation) to systems other than Phobos, such as the Saturn and Jupiter systems. Results. We present the variational equation for Phobos’ rotation employing the symbolic Maple computation software. The adjustment test simulations confirm the latitude libration of Phobos, suggesting gravity field coefficients obtained using a shape model and homogeneous density hypothesis should be re-examined in the future in the context of dynamics. Furthermore, the simulations with different k2 values indicate that it is difficult to determine k2 efficiently using the current data.

Dynamical analysis of exact optical soliton structures of the complex nonlinear Kuralay-II equation through computational simulation

In this paper, we successfully extracted the various types of soliton solutions for the complex nonlinear Kuralay-II equation through the improved F-expansion method with symbolic computational software Mathematica. The extracted soliton solutions for the Kuralay-II equation are interesting, novel and more general such as anti-kink wave solitons, dark solitons, kink wave solitons, bright solitons, periodic wave solitons, mixed solitons in bright-dark soliton shape, peakon solitons, and solitary wave structures. The graphical structure of some extracted solutions is visualized in 2D, 3D and contour plottings with imaginary, real, and absolute values of the functions by using the numerical simulation. The proposed research will contribute to advancing our knowledge about the complex nonlinear Kuralay-II equation and demonstrating the applicability to the proposed approach to investigate other higher-order complex nonlinear equations. The successful investigation demonstrated that the proposed method is effective, simple, more powerful, efficient and can be utilized on a variety of other nonlinear equations. The explored solitary waves and optical solitons will play an important role in the investigation of nonlinear phenomena in various domains of science and engineering.

Unraveling metachronal wave effects on heat and mass transfer in Non-Newtonian fluid

Non-Newtonian fluids flow generated by “cilia” are critical in medicine and bio-medical engineering. Such investigations are created by the back-and-forth movement of a microscopic hair-like structure connected to the walls, which causes a metachronal wave to form and drive biological fluids. Motivated by a wide range of biological applications, this study aims to explain the incompressible flow of Ellis fluid caused by the propagation of an infinite metachronal wave train traveling along channel walls owing to constantly beating cilia. The problem is simplified by low Reynolds number and long wavelength assumption. The mathematical model is solved with the aid of symbolic computational software Mathematica 13 version. The consequences of emerging parameters are then shown in graphical form and discussed comprehensively. From the study, it is worth mentioning that velocity declines with increasing material constants. Temperature distribution is also improved in the core sector of the channel and reduced at the walls. It is predicted that this approach will make an essential contribution to the progress and enhancement of various types of drug delivery systems in the biomedical industry and biomechanics.

Magnetic williamson hybrid nanofluid flow around an inclined stretching cylinder with joule heating in a porous medium

The study of an inclined stretching cylinder in a porous medium holds significant implications for understanding complex fluid dynamics and heat transfer phenomena, mainly when influenced by a hybrid nanofluid containing Cu and Al2O3, Joule heating, and a magnetic field in the presence of Williamson fluid. The nonlinear ordinary differential equations are derived by applying pertinent similarity transformations to partial differential equations. Subsequently, these non-dimensional ordinary differential equations are transformed into a system of first-order ODEs and numerically solved using Maplesoft's symbolic computation software MAPLE 2023. The influence of governing parameters on dimensionless velocity, temperature, concentration, skin friction, heat, and mass transfer rates is investigated. This investigation is crucial for various practical applications, including geothermal reservoirs, enhanced oil recovery, and environmental processes involving porous media. It is demonstrated that Biot and Eckert numbers and nanofluid parameters contribute to increased surface temperature, while the Lewis number leads to a decrease in dimensionless concentration. Moreover, the curvature parameter is associated with an increase in dimensionless concentration, and skin friction demonstrates a direct relationship with the magnetic parameter while inversely correlating with the cylinder's curvature.These results contribute to the scientific understanding of these complex interactions and provide valuable insights for engineering applications, such as designing efficient heat exchangers, optimizing cooling systems, and advancing technologies involving fluid flow and magnetic fields.

Weakly restoring forces and shallow water waves with dynamical analysis of periodic singular solitons structures to the nonlinear Kadomtsev–Petviashvili-modified equal width equation

In this study, the nonlinear two-dimensional Kadomtsev–Petviashvili-modified equal width (KP-MEW) equation is under investigation, which is described as a nonlinear wave model in weakly restoring forces and shallow water waves in the way of ferromagnetic, solitary waves in two dimensions with short amplitude and long wavelength and are also helpful in the investigation of various behaviors in nonlinear sciences. We successfully found the interesting and novel solitary wave results in bright solitons, kink solitons, dark solitons, anti-kink solitons, periodic singular solitons for nonlinear two-dimensional KP-MEW equation on the base of extension of simple equation technique under symbolic computational software Mathematica. The created solitons play an important role in nonlinear engineering and physics such as nonlinear optics, nonlinear dynamics, soliton wave theory, optical fiber and communication system. We are sure that these constructed results are novel and interesting which does not exist in the previous literature works. The graphical representation of constructed results is shown by 2D, 3D and contour graphs. The investigated research work proved that utilized technique is more concise, straightaway and efficient for study of other nonlinear partial differential equations.

A Novel Approximation Approach for the Analytical Solution of the Flow of Micropolar Fluid Through a Permeable Channel

An attempt is made in this study to investigate the problem of micropolar fluid flow in a porous medium theoretically. Employing the Berman’s similarity solution, the model equations governing the flow is transformed into a set of nonlinear ordinary differential equation and solved using Temimi-Ansari method. Expressions for the velocity and micro-rotation profiles are obtained under the impressions of diverse parameters affecting the flow problem. Using symbolic computation software Mathematica, the nondimensional equations are solved numerically using the Keller Box scheme. Comparison between the analytical solution obtained by TAM and the numerical result are compared with results in literature to observe rapid convergence. Findings from the study showed in the presence of

Spacelike Ac-Slant Curves with Non-Null Principal Normal in Minkowski 3-Space

In this paper, we define a spacelike ac-slant curve whose scalar product of its acceleration vector and a unit non-null fixed direction is a constant in Minkowski 3-space. Furthermore, we give a characterization depending on the curvatures of the spacelike ac-slant curve. After that, we get the relationship between a spacelike ac-slant curve and several distinct types of curves, such as spacelike Lorentzian spherical curves, spacelike helices, spacelike slant helices, and spacelike Salkowski curves, enhancing our understanding of its geometric properties in Minkowski 3-space. Finally, we used Mathematica, a symbolic computation software, to support the notions of an ac-slant curve with attractive images.

Multiple Soliton Solutions for Coupled Modified Korteweg–de Vries (mkdV) with a Time-Dependent Variable Coefficient

In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for coupled mKdV with a time-dependent variable coefficient. We implement the Hirota bilinear technique to compute analytical solutions for the coupled mKdV system. Such calculations are made by using a software with symbolic computation software, for instance, Maple. Recently some researchers used Maple in order to show that the bilinear method of Hirota is a straightforward technique which can be used in the approach of differential, nonlinear models. We analyzed whether the experiments proved that the procedure is effective and can be successfully used for many other mathematical models used in physics and engineering. The results of this study display that the profiles of multiple-kink and singular-kink soliton types can be efficiently controlled by selecting the particular form of a similar time variable. The changes in the solitons based on the changes in the arbitrary function of time allows for more applications of them in applied sciences.

The Lie algebra of the lowest transitively differential group of degree three *

We investigate the real Lie algebra of first-order differential operators with polynomial coefficients, which is subject to the following requirements. (1) The Lie algebra should admit a basis of differential operators with homogeneous polynomial coefficients of degree up to and including three. (2) The generator of the algebra must include the translation operators for all the variables . (3) The Lie algebra is the smallest indecomposable Lie algebra satisfying (1) and (2). It turns out to be a 39-dimensional Lie algebra in six variables (k = 6) and the construction of this algebra is also the simplest possible case in the general construction of the Lie algebras of the transitively differential groups introduced by Guillemin and Sternberg in 1964 involving the coefficients of degree 3. Those algebras and various subalgebras have similarities with algebras related to different applications in physics such as the Schrödinger, Conformal and Galilei algebras with and without central extension. The paper is devoted to the presentation of the structure and different decompositions of the Lie algebra under investigation. It is also devoted to the presentation of relevant Lie subalgebras and the construction of their Casimir invariants using different methods. We rely, in particular, on differential operator realizations, symbolic computation packages, the Berezin bracket and virtual copies of the Lie algebras.

Numerical and analytical solutions of new Blasius equation for turbulent flow

The Blasius equation for laminar flow comes from the Prandtl boundary layer equations. In this article, we establish a new and generic Blasius equation for turbulent flow derived from the turbulent boundary layer equation that can be used for turbulent as well as laminar flow. The analytical and numerical solutions have been investigated under specific conditions to the developed new Blasius equation. The analytical and numerical results have been compared through tables and graphs to validate the established model. In fluid dynamics, analytical solutions to complicated systems are tedious and time-consuming. Changing one or more constraints can introduce new challenges. In this case, symbolic computation software provides an easier and more flexible solution for fluid dynamical systems, even if boundary conditions are adjusted to explain reality. Therefore, the MATLAB code is used to investigate the new third-order Blasius equation. The comparison and graphical representations demonstrate that the achieved results are encouraging.

Bifurcation analysis, stationary optical solitons and exact solutions for generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and quintuple power-law of refractive index in optical fibers

In this paper, the bifurcation, stationary optical solitons and exact solutions in optical fiber propagation are studied. The propagation model is described by a generalized nonlinear Schrödinger equation with Kudryashov’s quintuple power law of refractive index. First, the equation is transformed into a second-order nonlinear ordinary differential equation by an appropriate transformation. Then, the bifurcation of the phase diagram of the obtained ordinary differential equation is established using planar dynamic system theory, and the types of optical solitons and exact solutions are determined. What is more, the stationary optical solitons and exact solutions are obtained by the polynomial complete discriminant system, and are demonstrated in 3D and 2D graphics by using the symbolic computation software Maple. Finally, the obtained stationary optical solitons and exact solutions include the Jacobi elliptic function solution, hyperbolic function solution, rational function solution, trigonometric function solution and exponential function solution. Moreover, the results provide a method to further study stationary optical solitons and exact solutions of propagation pulses in optical fibers.

Increasing beam stability zone in synchrotron light sources using polynomial quasi-invariants

The objective of this article is to propose a scheme to increase the stability zone of a charged particles beam in synchrotrons using a suitable objective function that, when optimized, inhibits the resonances onset in phase space and the dynamic aperture of electrons in storage rings can be improved. The proposed technique is implemented by constructing a quasi-invariant in a neighborhood of the origin of the phase space, then, by using symbolic computation software, sets of coupled differential equations for functions involved in nonlinear dynamics are obtained and solved numerically with periodic boundary conditions. The objective function is constructed by proposing that the innermost momentum solution branch of the polynomial quasi-invariant approaches to the corresponding ellipse of the linear dynamics. The objective function is optimized using a genetic algorithm, allowing the dynamic aperture to be increased. The quality of results obtained with this scheme are compared with particle tracking simulations performed with available software in the field, showing good agreement. The scheme is applied to a synchrotron light source model that can be classified as third generation due to its emittance.

AN ASYMPTOTIC ELECTROSTATIC MODEL OF AN ARRAY OF MICROMIRRORS

This paper discusses a multiscale electrostatic model of a two-dimensional micromirror array. It is applicable to very large arrays with several zones of electrical actuation. The model is made with periodic solutions and four kinds of boundary layer effects at outer boundaries, interfaces between different actuation zones, and outer and inner edges. This work is done in the context of the development of a symbolic calculation software based on an extension-combination principle, so that the model derivations are constructed in such a way as to follow the same algorithm.

On the investigation of optical soliton solutions of cubic–quartic Fokas–Lenells and Schrödinger–Hirota equations

Purpose:When it comes to third and higher-order dispersion, the Schrödinger–Hirota equation is one of the main models developed outside the classical NLSE management models for optical soliton transmission. The cubic–quartic Fokas–Lenells equation is also one of the recently developed equations, which has importance in the field of telecommunications regarding optical soliton transmission in the absence of chromatic dispersion. In this study, in order to examine the optical solitons, the Schrödinger–Hirota equation in the presence of the chromatic dispersion and the cubic–quartic Fokas–Lenells equation discarding the chromatic dispersion were investigated. For this intent, by obtaining certain soliton types using the unified Riccati equation expansion method (UREEM), optical soliton solutions were obtained for both models and graphical representations and comments were made. Methodology:By developing appropriate computer algorithms and applying UREEM in the following ways, symbolic calculation software was made and analytical optical soliton solutions were obtained. Findings:Through computer algebra software, we plotted the obtained results via 3D, 2D views and we also illustrated the investigation of wave behavior caused by parameter change on 2D graphics. Originality:Different soliton behavior under the parameters effect of the Schrödinger–Hirota equation having chromatic dispersion and the cubic–quartic Fokas-Lenells equation is investigated and the obtained results are reported.

Exact solutions of the stochastic fractional long–short wave interaction system with multiplicative noise in generalized elastic medium

The long–short wave interaction system (L-SWIS) is an important model describing the interaction of two waves propagating in a generalized elastic medium. In this study, the bifurcation and the influence of random interaction on the exact solution of the stochastic fractional long–short wave interaction system (SFL-SWIS) with multiplicative Brownian motion are studied, where the derivative refers to the modified Riemann–Liouville definition. After the Hamiltonian system is established by traveling wave transformation and first-order integration, we obtain abundant exact parametric solutions of SFL-SWIS. Also, the three-dimensional graph of some obtained results are drawn by using the symbolic calculation software Maple, which is used to show the influence of fractional derivative and random interaction on the solution. The obtained solitary wave solutions and exact solutions include the elliptic function solution, trigonometric function solution, exponential function solution and singular periodic solution. This study is the first to investigate the bifurcation and exact solution of SFL-SWIS, and the obtained results are theoretically and practically important to the propagation of optical signals, plasma waves and ocean waves.

The exact solutions of the [formula omitted]–dimensional Kadomtsev–Petviashvili equation with variable coefficients by extended generalized [formula omitted]-expansion method

In this paper 2+1–dimensional Kadomtsev–Petviashvili (KP) equation with variable coefficients is investigated through the extended generalized G′G–expansion technique. One of the most universal model is KP equation, which is used to explain the ion acoustic waves in plasma physics, to model two dimensional shallow water waves, and in ferromagnetic, Bose–Einstein condensation and string theory. The obtained exact solutions of KP equation are in the form of hyperbolic function, trigonometric function, and rational function. With the aid of symbolic computational software Mathematica, the three dimensional surface plots with corresponding contour plots are provided for the obtained closed from solutions, which are of the form of solitary waves, multi solitons and periodic solitary wave like dynamical structures.

Exploration of radiative heat on the Jeffery fluid flow in a microchannel for the impact of suction/injection

AbstractThe purpose of the present paper is to investigate the flow and heat transfer of thermal radiation on the Jeffery fluid flow within a microchannel for the effects of the superhydrophobic surface (SHS) within suction/injection. The governing differential equations of motion and heat transfer are transformed into nonlinear coupled ordinary differential equations (ODEs) using appropriate similarity transformations. The ODEs are solved along with boundary conditions by adopting Runge–Kutta with shooting technique. Symbolic computational software such as MATLAB, the solver bvp4c syntax examines the behavior of the relevant physical parameters. However, some effective emerging parameters on the flow problem reveal that the microchannel walls within the suction/injection parameter increase the temperature, and the SHS is heated. In contrast, without slip, the opposite behavior is rendered. It is clearly shown that the velocity profile diminishes with increasing the Prandtl number. Furthermore, it is noticed that velocity decreases for increasing values of Hartmann number. Comparison with available results for particular cases is an excellent agreement.

New exact solutions of space-time fractional Schr¨odinger-Hirota equation

In this study, improved Bernoulli sub-equation function method (IBSEFM) is presented to construct the exact solutions of the nonlinear conformable fractional Schr¨odinger-Hirota equation (FSHE). By using the traveling wave transformation FSHE turns into the ordinary differential equation (ODE) and by the aid of symbolic calculation software, new exact solutions are obtained. 2D, 3D figures and contour surfaces acquired from the values of the solutions are plotted. The results show that the proposed method is powerful, effective and straightforward for formulating new solutions to various types of nonlinear fractional partial differential equations in applied sciences.